We determine the border ranks of tensors that could potentially advance the known upper bound for the exponent \(\omega \) of matrix multiplication. The Kronecker square of the small \(q=2\) Coppersmith–Winograd tensor equals the \(3\times 3\) permanent, and could potentially be used to show \(\omega =2\). We prove the negative result for complexity theory that its border rank is 16, resolving a longstanding problem. Regarding its \(q=4\) skew cousin in \({\mathbb {C}}^5{\mathord { \otimes } }{\mathbb {C}}^5{\mathord { \otimes } }{\mathbb {C}}^5\), which could potentially be used to prove \(\le 2.11\), we show the border rank of its Kronecker square is at most 42, a remarkable sub-multiplicativity result, as the square of its border rank is 64. We also determine moduli spaces VSP for the small Coppersmith–Winograd tensors.

我们确定了张量的边界等级，这些边界等级可能会提高矩阵乘法的指数\(\omega \)的已知上限。小的\(q=2\) Coppersmith–Winograd 张量的 Kronecker 平方等于\(3\times 3\)永久量，并且可能用于显示\(\omega =2\)。我们证明了复杂性理论的否定结果，它的边界等级是 16，解决了一个长期存在的问题。关于它在\({\mathbb {C}}^5{\ mathord { \otimes } }{\mathbb {C}}^5{\mathord { \otimes } }{\ mathbb {C}}^5\)，它可能被用来证明\(\le 2.11\)，我们显示其 Kronecker 平方的边界秩最多为 42，这是一个显着的子乘法结果，因为其边界秩的平方为 64。我们还确定了小 Coppersmith-Winograd 张量的模空间 VSP。